set your lowest allowable shutter
That would certainly fix my problem. I had posted this exact question last year about the t3i but there was no function for it. I'll search for the 6D.
No offense, but it seems to me that you are locked in the mindset where aperture and shutter speed only are means of aesthetic control.
I am. No shame about it. I bought a Full Frame cam that have excellent ISO performance to use it. Auto-ISO is a god send for someone who want to keep tight control on the aperture and speed while keep a fast shooting rate in changing lighting. Just to say, the t2i, t3i, t4i all have this function. You can compensate in manual mode no problem. I think it's just something that they forgot in the bigger models because the techs didn't use it.
I will of course follow your advice and try the AF and TV mode. If I can indeed put a minimum shutter speed, that will help. Would be even better if I could do it per lens.
What type of shooting are you doing?
Street photography, often low light. Shallow DoF. 35mm, 85mm, 135mm. I always prefer to stay over 1/300 for my shutter speed. I got a lot more keepers that way. I dont want to look at my metering each time I take a photo. I need to concentrate on the surrounding.
Actually the cam do it nicely. I just want to compensate when there's a backlight or some strong sun.